confidence inlerval in the original scale. When 

 there are zero observations, however, the data 

 must be rescaled prior to log-transformation (by 

 adding some arbitrary positive value to each ob- 

 servation) and this causes the geometric mean 

 and its confidence interval to become biased. Al- 

 though no studies have been reported investigating 

 the effect of zero observations on the geometric 

 mean, it seems reasonable to assume that the bias 

 will depend on the choice of the constant added 

 to each observation and that it will increase with 

 the proportion of zeros in the data. 



Numerical simulation: The basic premise in 

 this simulation was that, for data with an approx- 

 unate delta-distribution, the h-mean mean and the 

 sample median were the only correct statistics to 

 estimate the population mean and the mid-point 

 of the distribution, respectively. On the basis of 

 distribution theory it was already known that the 

 sample mean and its variance were biased estima- 

 tors of the lognormal population mean and its 

 variance, and that the geometric mean was a bi- 

 ased estimator of the middle point of lognormal 



Table 1. Typical sample size, proportion of zero observations, and variability for 

 data collections from NUEL programs where 5-means have been used. 



data with many zeros. Therefore, the purpose of 

 the simulation was to describe the relationships 

 among the four statistics (i.e., their relative loca- 

 tions) and to investigate how their magnitudes 

 and the width of their 95% confidence intervals 

 (95%C/) were affected by the presence of zeros 

 in the data and by different amounts of variability. 



The data base used in the numerical simulation 

 consisted of three data sets of 100 normal random 

 numbers with identical mean (3c = 2.00) and in- 

 creasing variances {s = 0.50, 1.00, and 2.25) so 

 that the CV's would be 25, 50, and 75 %. These 

 data were converted into lognormally distributed 

 data by exponentiation of each observation. 

 Therefore, the three lognormal data sets had iden- 

 tical geometric mean, GM = exp(2) = 7.39, and 

 increasing variabilities: CV = 25%, 50%, and 

 75% in the logarithmic scale. These CV values 

 roughly corresponded to typical (low, moderate, 

 and high) variabilities encountered in samples of 

 marine organisms in various monitoring programs 

 at NIJCL (Table 1). 



Monitoring Program 



Sample Size (n) 



Proportion 

 of Zeros (5) 



Variability (CV)' 



I'ish trawl surveys 



Ichthyoplankton surveys 



Lx)bster larvae entrainment 



Rocky Shore (% cover) 



Coefficient of variability for log-transformed data. 



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